Logic alone does not a system make. Boolean equations provide the means to transform a set of inputs into deterministic results. However, these equations have no ability to store the results of previous calculations upon which new calculations can be made. The preceding adder logic continually recalculates the sum of two inputs. If either input is removed from the circuit, the sum disappears as well. A series of numbers that arrive one at a time cannot be summed, because the adder has no means of storing a running total. Digital systems operate by maintaining state to advance through sequential steps in an algorithm. State is the system’s ability to keep a record of its progress in a particular sequence of operations. A system’s state can be as simple as a counter or an accumulated sum. State-full logic elements called flip-flops are able to indefinitely hold a specific state (0 or 1) until a new state is explicitly loaded into them. Flip-flops load a new state when triggered by the transition of an input clock. A clock is a repetitive binary signal with a defined period that is composed of 0 and 1 phases as shown in Fig. 1.10. In addition to a defined period, a clock also has a certain duty cycle, the ratio of the duration of its 0 and 1 phases to the overall period. An ideal clock has a 50/50 duty cycle, indicating that its period is divided evenly between the two states. Clocks regulate the operation of a digital system by allowing time for new results to be calculated by logic gates and then capturing the results in flip-flops. There are several types of flip-flops,
